Is there no translation for (logical) "argument" ?

[Elizabeth]

I don't think so. Here is the appropriate Merriam-Webster's definition for "ground(s)" :

a basis for belief, action, or argument <ground for complaint> -- often used in plural b (1) : a fundamental logical condition (2) : a basic metaphysical cause

The "ground(s)" is the basis for the argument, but not the argument itself.

It may help to think of "Argument" as the debate over capital punishment itself, the "grounds" being particular reasons (arguments 1-5, if you will). On the grounds of, on the grounds that, grounded in....then becomes 根拠 or 理由 or whatever in each particular case. Clearly these concepts are translatable, and distinctive enough to have their own set of words associated.